Please try clearing your game cache, and see if this has any effect on things. To do this, close the game and navigate to the main game folder, then the swtor folder. Delete the DiskCacheArena file, then restart the game.

If the issue persists, have you tried resetting your keybindings on an affected character? It may be a bit of a pain to re-set them to how you like, but it'll help to check if somehow they're out of whack.

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